14.8 - Lagrange Multipliers

Consider the problem of maximizing or minimizing some function $f(x,y)$, subject to some constraint $g(x,y)=k$

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Important

$g(x,y)=k$ is a contour/level curve of some unknown constraint surface $z=f(x,y)$.

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Solving This Problem

1. We parameterize the constraint curve $g(x,y)=k$ by

$\vec{r}(t)=⟨x(t),y(t)⟩$

By the Key Formula from Sec. 14.6,

$\nabla g\cdot {\vec{r}}^{\prime }(t)=0$

for all t in the parameter domain.

2. The curve C whose optimal value we seek is obtained by lifting $\vec{r}(t)$ to the surface $z=f(x,y)$.

This means we can parameterize C by

$\vec{c}(t)=⟨x(t),y(t),f(x(t),y(t))⟩$

Observe

The max/min values along $\vec{c}$ may occur when the z-component of ${\vec{c}}^{\prime }s$ tangent vector equals 0.

Why?

In complete analogy with Calc 1, we want to look where $\vec{c}$ has a horizontal tangent line. In ${\mathbb{R}}^3$